investigation of the scalability of dynamic wavelength-routed optical networks [invited]

Investigation of the scalability of dynamicwavelength-routed optical networks [Invited]

Michael Düser, Alejandra Zapata, and Polina Bayvel

Optical Networks Group, Department of Electronic and Electrical Engineering,University College London (UCL), London WC1E 7JE, UK

[email protected]; [email protected]; [email protected]

http://www.ee.ucl.ac.uk/~ong/

RECEIVED 2 JUNE 2004;REVISED 5 JULY 2004;ACCEPTED7 JULY 2004;PUBLISHED 6 AUGUST 2004

We describe results of the scalability analysis for dynamic wavelength-routed op-tical networks with end-to-end lightpath assignment and central network controlwith electronic scheduling and processing of lightpath requests. We investigatethe effect of the algorithm complexity in both the scheduling and the dynamicrouting and wavelength assignment (DRWA) of lightpath requests. Schedulingtheory and static performance-prediction techniques were applied to define thebounds on the electronic processing time of requests, and hence the maximumnumber of nodes supported by a centralized dynamic optical network for givenblocking probability, latency, and network diameter. Scalability analysis resultsshow that medium-sized centralized networks (∼50 nodes) can be supportedwhen these networks are reconfigured on a burst-by-burst basis. In addition,we found that real topologies showed a complex trade-off between the requestprocessing time, blocking probability, and resource requirements. These findingscan be used to determine the optimum combination of scheduling/DRWAalgorithm, showing that the fastest DRWA algorithm does not necessarily leadto the minimum blocking probability and maximum scalability but that a carefulconsideration of both blocking and processing speed is required. The resultsare applicable both to dynamic network architectures with centralized requestprocessing such as wavelength-routed optical networks and to the design ofadvanced optical switching matrices and routers. © 2004 Optical Society ofAmerica

OCIS codes:060.0060, 060.2330, 060.4250, 060.4510.

1. Introduction

Although static wavelength-routed optical networks (WRONs) [1] are simple to design andimplement, they require significant wavelength overprovisioning in order to accommodatevarying input traffic and to be resilient to failures [2, 3]. Dynamic allocation of lightpathshas been proposed as an alternative to provide the same service but with reduced resources.

Although dynamic networks can be implemented in a centralized or distributed way,centralized provisioning is attractive because the central control node maintains global in-formation on the network state (topology and wavelength utilization) [4]. This achieves amore efficient allocation of resources and, therefore, a lower blocking probability than dis-tributed dynamic routing and wavelength-assignment (DRWA) algorithms (e.g., Ref. [5]).However, centralized systems have the potential risk of poor survivability and scalability,which might render them impractical. Survivability (i.e., the ability of the network to sur-vive failures) is improved by redundancy of the control information in one (or more) backupcontrol nodes. Therefore, on possible failure of the main control node, network operation is

© 2004 Optical Society of AmericaJON 4484 September 2004 / Vol. 3, No. 9 / JOURNAL OF OPTICAL NETWORKING 674

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not disrupted, as the control of the network is transferred to the backup node. For an exam-ple of a real long-haul centralized optical network utilizing control node redundancy (seeRef. [6]). Scalability (i.e., the maximum number of nodes that can be supported by suchdynamic optical network architectures), however, remains a fundamental drawback of cen-tralized networks, as a single node must maintain all the information on the network stateand perform the processing of all the lightpath requests generated by the network nodes.With the number of increasing network nodes (or edge routers) all generating requests, thekey question is, How scalable is the centralized implementation?

The scalability is limited by the processing capacity in the central node, the compu-tational complexity of the DRWA algorithm, and the quality-of-service (QoS) constraints(latency and blocking). Increased complexity translates into longer processing times of re-quests arriving at the control node, limiting network scalability. Yet all DRWA algorithmsproposed to date experience a trade-off between computational complexity and blocking, aswe show below. In this paper we analyze two DRWA algorithms, namely, the shortest-pathfirst-fit (SP-FF) [7] and adaptive unconstrained routing—exhaustive (AUR-E) [8], which todate define the lowest and highest computational complexity, respectively.

The low computational complexity of the SP-FF algorithm is achieved, first, by solv-ing separately the routing and the wavelength-assignment problems (as opposed to con-currently searching for a route and a wavelength) and, second, by choosing the fastestalgorithms to solve each of the mentioned tasks. Fastest routing is achieved by using off-line precomputed shortest paths, with a computational complexity of routing table lookupequal toO(N), whereN is number of nodes, assuming each edge router is attached to acore node. The first-fit wavelength-allocation scheme has been shown to have the lowestcomputational complexity (computational complexityO(W), W, number of wavelengths)among the proposed wavelength-allocation algorithms to date [8, 9] (actually, in these ref-erences random fit has the same computational complexity of first-fit, because the extracomputations required for executing the random experiment were not considered).

Conversely, the high computational complexity of the AUR-E algorithm exists becausethe routing and wavelength-allocation problem is solved concurrently, by executing theDijkstra algorithm online as many times as the number of wavelengths with which the net-work links are equipped [computational complexityO

(WN2

)when the Dijkstra algorithm

is implemented with static arrays].Although the SP-FF algorithm has been widely used because of its simplicity and the

good performance of the FF heuristic alone [10–12], it results in much higher blockingprobability than AUR-E when restricted to shortest-path (SP) search (more than two ordersof magnitude were shown in Refs. [8, 13]). This is because the routing part (here, SP) hasa much higher effect on the blocking performance than the wavelength assignment (here,FF), as shown in Ref. [8]. However, the better performance of the AUR-E algorithm (shownto be the best performing algorithm to date [8, 13–15]) comes at the expense of complexcomputations involving the Dijkstra algorithm, decreasing scalability.

In this paper, we investigate how to maximize the processing speed through parallelprocessing of lightpath requests to increase network scalability. Both the SP-FF and AUR-E algorithms offer the possibility of achieving this, and it is shown for the first time howthe electronic processing, in particular the problem of memory lookups, can be optimizedto achieve maximum network scalability. In addition, a new implementation for the SP-FFalgorithm is proposed that decreases its computational complexity fromO

(N3)

to O(N2),

further increasing its scalability.Earlier analysis of DRWA algorithms in the context of optical-burst-switched (OBS)

networks has provided only partial understanding of the scalability problem in dynamicWRONs. In Ref. [16] the maximum number of nodes supported by a centralized networkwas quantified for different DRWA algorithms for generic topologies, but QoS require-


ments were not taken into account. In Ref. [17] the effect of QoS requirements on scalabil-ity was analyzed, but the processing times of specific DRWA algorithms were not consid-ered, nor were the results applied to practical topologies that introduce differential delays,potentially affecting the fairness of the lightpath processing. Although in Ref. [18] QoSrequirements and practical topologies were taken into account, only asymptotic complex-ity analysis was carried out to estimate the processing time of a single class of DRWAalgorithms, which did not include the investigation of the best- and worst-case scenarios.

In this paper we present, to the best of our knowledge, for the first time, the analysisof the scalability of dynamic WRONs considering different types of DRWA algorithms ap-plied to practical physical network topologies. A key parameter that has been included inour calculations is the propagation (round-trip) time between the edge and control node,not considered previously, which limits the network scalability in combination with QoSconstraints such as latency and blocking probability. Our results show that medium-sizenetworks of up to 50 nodes (exceeding the largest continental network studied here) can beeasily deployed with centralized architectures and that the fastest algorithm may be subop-timal for implementation, given its increased capacity requirement to yield an acceptableblocking performance. Although the research is mainly focused on the understanding ofthe scalability of dynamic optical networks, the results also apply to the design of opticalswitching matrices within optical routers [19].

This paper is organized as follows: The network architecture, the input traffic statistics,and the request scheduling in the central control node are described in Section2, whichgives the mathematical description of the scalability as a function of the request processingtime. Section3 is focused on quantifying the request processing time for different DRWAalgorithms, and the electronic implementation in state-of-the-art routers, considering thecontributions of the arithmetic-logical unit (ALU) speed and the memory access time. InSection4, these results are applied to seven real network topologies, to answer two keyquestions: (a) how scalable dynamic architectures are as a function of the network topol-ogy, given QoS constraints, DRWA employed, and its electronic implementation; and (b)whether dynamic networks can live up to the promise of achieving significant resourcesavings as compared with static WRONs and, therefore, whether scalability would increasegiven that a smaller wavelength space would be searched by the DRWA algorithm.

2. Network Architecture and Request Scheduling

Figure1 shows the schematic for a centralized dynamic network architecture. It consistsof an optical core of switches, each connected locally to an edge router where incomingpackets are classified according to destination and QoS requirements. A key assumptionin our study is that the statistical processes describing incoming traffic streams can bearbitrary. This is based on (our) previous results on the design of adaptive traffic aggregatorsat the network edge, where it was shown that it is possible to generate smooth lightpathrequest processes (i.e., a coefficient of variation [20] lower than for Poisson arrival) forarbitrary input traffic processes [21]. Every edge router is equipped with a buffer for burstaggregation per destination/QoS pair. During the burst aggregation process the lightpathrequest state is triggered once predefined QoS parameters (e.g., latency or burst size) aremet, and a request is sent to the control node. The processing of requests in the controlnode is divided into two tasks [Fig.2(a)]:

1. Schedulingof requests according to their assigned class of service (CoS), and thepropagation delay between edge routers and control node. The algorithms used andtheir applicability to calculate the network scalability are discussed next.

2. Processingof lightpath requests using one of a variety of DRWA algorithms. Tospeed up the serial processing of lightpath requests, parallel electronic processing


can be used to carry out some parts of the lightpath-assignment process [Fig.2(b)].The achievable speed-up is investigated in Section3. In the course of this study,the SP-FF and the AUR-E algorithms were used, since the first represents the caseof fast processing with high blocking, whereas the latter achieves low blocking butexperiences long processing times. In addition, an intermediate solution (namely,k-SP-FF, which explores the firstk disjoint shortest paths) was considered for veri-fying whether similar performance to the AUR-E algorithm can be achieved at lowercomputational complexity.

ACK

Request scheduler

DRWAAlgorithmrequests

Core router

Edge router

CONTROL NODE

request

Dynamic lightpath

Highest priority queue

Lowest priority queue

Fig. 1. Dynamic optical network with centralized lightpath allocation (ACK, acknowledge-ment of lightpath request).

At the control node, requests are assigned priorities according to criteria such as CoSand distance from the control node. Then, a requestschedulingalgorithm selects the nextrequest to be processed by the DRWA algorithm applying fairness rules by taking into ac-count the nonnegligible propagation delay of requests between the edge and the controlnode. These propagation delays can reach significant values of several milliseconds forwide-area networks (WANs), resulting in the unfair treatment of nodes farthest from thecentral node. If the DRWA algorithm is successful in finding a lightpath, an acknowledge-ment is sent to the corresponding source node, and the network is configured to establishthe lightpath. Otherwise, the request is dropped with a no-ACK message sent to the sourcenode. The following scheduling algorithms were investigated:

• First-in/first-out (FIFO). In the simplest case there is no scheduling, with requestsprocessed in the same order as they arrive at the control node.

• Rate monotonic (RM).The RM algorithm was originally designed for the schedul-ing of several periodic and time-critical events by a single microprocessor, e.g., incontrol engineering [22]. Priority for a requestρi is assigned according to the periodTi (the time between successive arrivals of requestρi). Requests with shortestTi havethe highest priority. As quantified below, the RM algorithm is less efficient than theearliest deadline first (EDF) algorithm described next, but unlike the EDF, the RMalgorithm can provide service guarantees even in transient overload situations [22].


-33-

µPµP µPµP µPµP

1 2 k

network-widerequest arrival

Comparison

RT RT RT

Optimum result

Scheduling

DRWA

Figure 2

(a) single processor

µPµProuting

table


R/W

DRWA RT

Optimum result

Scheduling

(b) parallel processing for k-SP-FF (left) and AUR-E (right)

µPµP µPµP µPµP1 Nλ


Comparison

RT RT RT

Optimum result

2

Scheduling

DRWA

Fig. 2. Scheduling and request processing (a) for single-processor architecture (b) formulti-processor architecture with (left) parallel processing ofk precomputed shortest pathsfor thek-SP-FF algorithm and (right) parallel processing for all wavelengths for the AUR-Ealgorithm.

• Earliest deadline first (EDF). Every request has a field specifying a deadline bywhen it must be processed [23]. The request with the earliest deadline is assigned thehighest priority; hence it also works for nonperiodic request arrivals, i.e., providesthe highest flexibility, and also has the highest scheduling efficiency.

FIFO scheduling is suitable for best-effort networks (without QoS constraints). RMand EDF are best suited for networks with strict delay requirements, since they guaranteean incoming lightpath requestρik (i, CoS;k, source node) to be processed within a givendeadline, providing that the processor utilization is below a boundU [24], whereU (0≤U ≤ 1) is the (dimensionless) processor utilization per requestρik, and depends only on thefollowing:

• The processing time per request,C. A detailed description of how to calculateC fordifferent DRWA algorithms is given in Section3.

• The periodicity of request arrivals at the processor in the central control node,Tik.


For FIFO schedulingTik ≡ tedge,ik, wheretedge,ik is the maximum time that data atthe source edge node is held before transmission through the optical core (referredto as the edge delay [17]). For EDF and RM scheduling,Tik ≡ tedge,ik only for shortdistances between the edge and control nodes, e.g., in a local-area network (LAN).For WANs, however, it is vital to modifyTik, to consider the round-trip time delay(tRTT,ik, the time to propagate the request to the central node and return the acknowl-edgement to the source node). In conventional schedulingTik represents the processortime available for a particular request, since the round-trip time between the proces-sor and the originator of the request is negligible. This is not true, however, in WANs,where a request needs to propagate to the control node, is processed, and an acknowl-edgement returned. The total timeTik between two consecutive requests must, hence,be reduced bytRTT,ik, so that for the scheduler the periodicity of requests appears tobe reduced. This will ensure that requests from farthest nodes from the control nodehave higher priority than requests from the nearest nodes, so the delay experiencedby the different network nodes is equalized. For nonnegligibletRTT the time availablefor processing decreases, and hence the periodicity,Tik becomes

Tik = tedge,ik− tRTT,ik. (1)

For a network to be able to process all the requests in finite time (FIFO case) or beforea given deadline (RM and EDF), the following condition must be satisfied:

NCoS

∑i=1

N

∑k=1

(N−1)Cik

Tik≤U, (2)

whereNCoS is the number of classes of services (1 in the case of FIFO scheduler),N thenumber of nodes. The limits for the three different scheduling algorithms under considera-tion are as follows [24]:

NCoS

∑i=1

N

∑k=1

(N−1)Cik

Tik= U 6

{Ntot(21/Ntot−1)≤ 1 for RM1 for EDF and FIFO

, (3)

whereNtot is defined asNtot = NCoS×N(N−1) and it is the total number of sources gen-erating requests in the network. Further, for the limit of the RM algorithm:

limN→∞

URM = ln2≈ 0.69, (4)

which is the lower bound forURM and is used as a conservative estimate of the limit for theRM algorithm. With these limits onU , all requests can always be scheduled.

The physical network topology will also affect scalability, and it is discussed next. Inan idealized, starlike network architecture (Fig.3), all edge nodes are located the samedistance from the central control node, which allows us to simplify Eq. (3). For the sameCoS, the propagation delay to the control node is the same for all nodes; henceTik = Ti

andtRTT,ik = tRTT. Nmax is the maximum number of nodes for which the system is stable(no violation of latencies) for a given number of CoS,NCoS. It is further assumed that theround-trip times of all connections are identical (equidistant node spacing) and that theedge delay for every CoS is unique, leading totedge,ik = tedge,i . When we solve the quadraticequation given by Eq. (3), Nmax is given as

Nmax =

12

+

14

+U

(C

NCoS

∑i=1

1Ti

)−11/2

. (5)


edgerouter

switch

controlnode

Fig. 3. Optical network architecture in star topology, with edge nodes being equidistantfrom the control node.

For largeN, N(N−1)∼= N2, so that Eq. (5) can be simplified to

Nmax≈

√√√√U

(C

NCoS

∑i=1

1Ti

)−1 ∝

1√C

. (6)

Figure4 shows the maximum number of edge routers as a function of processing timeCassumed to be in the range 0.1–10µs (the reason for this range of values becomes clear inSection3), for the case of 3 CoS under RM and EDF schedulers, and 1 CoS (lowest latencyT1 only) with the EDF and FIFO schedulers. To ensure that network resources are usedefficiently, the data transmission time must be at least as long as the time required for settingthe lightpath (mainly determined by the round-trip timetRTT). This means that data shouldbe aggregated at the edge node at least fortRTT, which determines the minimum periodbetween consecutive lightpath requests. The following values were used:tRTT = 5 ms andrequest periodsT1 = 5 ms,T2 = 15 ms, andT3 = 45 ms. It can be seen that the numberof allowable edge routers decreases∝ C−1/2. ForC = 0.1 µs (equivalent to 100 cycles ofa 1-GHz processor) and 3 CoS, the network can support requests from up to 186 edgerouters without missing a deadline. When only one CoS withT1 = 5 ms implemented, thisincreases to 223. This implies, as expected, that the most time-sensitive requests (highestCoS) determine the overall network performance; additional CoS with less stringent delayrequirements can coexist with a minimal reduction in the allowable number of edge routers.Given that the scalability reduces∝ C−1/2, it is important to quantify its value accurately(which is not possible with asymptotic computational complexity analysis). In Section3we investigate the processing timeC as a function of the network topology, necessary forthe scalability analysis presented in Section4.

3. Request Processing Times

Because the time required for request processing is determined mainly by the speed of theDRWA algorithm, fast algorithms must be used for maximum scalability (Fig.4). This canbe achieved by minimizing online processing, which is usually done with precomputedroutes without checking the network status (topology and wavelength availability) for eachrequest. However, this leads to a higher blocking probability than in more computationallycomplex DRWA algorithms that take into account the network status to find a lightpath.This highlights an inevitable trade-off between scalability and blocking probability, which


0.1 1 100

100

200

No.

of e

dge

rout

ers

processing time per lightpath request [µs]

RM 3 CoS EDF 3 CoS EDF 1 CoS (high-priority) only

Fig. 4. Number of edge routers as a function of processing timeC and for 3 CoS using theRM or EDF scheduling algorithm (T1 = 5 ms,T2 = 15 ms,T3 = 45 ms), as well as 1 CoS(T1 = 5 ms) only for the EDF (and FIFO) algorithm. All calculations assumed a networkdiameter of 1000 km (tRTT = 5 ms).

is a problem that has not been investigated previously, but it is key for the practical imple-mentation of dynamic networks.

Among the large number of DRWA proposed to date (see Refs. [9, 25] for surveys), inthis paper we have focused on two, namely, theshortest-path first-fit(SP-FF) andadaptiveunconstrained routing-exhaustive(AUR-E) algorithms [8]. Both have been selected be-cause they represent two extreme (in terms of computational complexity, Section1) DRWAalgorithms, so best- and worst-case scalability can be evaluated. In addition, it was provedthat AUR-E yields the lowest blocking probability among all the proposed DRWA to date[8, 13, 15].

3.A. SP-FF Algorithm

The SP-FF algorithm was first introduced in Ref. [7]. It arbitrarily assigns integer numbers(indices) to wavelengths, and it selects the wavelength with the lowest index available inall the links of the shortest path between the source and destination. The implementationconsidered in this study is as follows. Let SPsd =

{lsd1, lsd2, . . . , lsd|SPsd|

}be the set of links

that make up the shortest path between source (s) and destination (d) nodes, computedoffline and stored for subsequent use. LetW = (W1,W2, . . . ,WL) be a vector ofL elements(L, number of links), where every element comprises|W| bits. The jth bit of elementWi

represents the availability of wavelengthj in link i (0 if it is idle and 1 otherwise). Uponrequest reception for a lightpath between nodess andd, the SP-FF algorithm executes thefollowing operations:

set_available_wavelengths=0 wavelength=-1 for l∈ SPsd

set_available_wavelengths=(set_available_wavelengths) BITWISE OR (W[l]) for i=1,2,�|W| if ((set_available_wavelengths BITWISE AND 2i) ==0) then

{ wavelength=i set_available_wavelengths=set_available_wavelengths BITWISE OR 2i break

} ACK(wavelength, SPsd) sent to source node

In the pseudocode above, abitwiselogical operation is a standard function of high-levelprogramming languages that performs a logical operation (AND, OR, etc.) between two


numbers by applying the logical operation to the corresponding bits of each number in asingle ALU operation (thus, bits are processed in parallel). Thus, all the wavelengths of thelinks of the path are processed simultaneously, leading to an asymptotic time complexityfor the lightpath search task ofO(L+W), whereL is the number of links andW the numberof wavelengths. Given thatL scales asO(N)—N number of network nodes—andW doesasO

(N2), the overall complexity results inO

(N2).

The scaling ofL was estimated considering that real networks have an average nodaldegreeδ (δ = 2L/N) between 2 and 5. Therefore,L = δN/2∼O(N). The scaling ofW wasobtained using the formula presented in Ref. [1] to estimate the minimum required numberof wavelengths in a static network, given byN(N−1)H/2L, whereH is the average pathlength (in number of hops). AsH andL scale withN, W scales withN2.

A complexity of O(N2)

is a significant reduction with respect to the previously pub-lished implementations of the SP-FF algorithm, which achieve a computational complex-ity of O(LW)[26], i.e., O

(N3). The decrease in the time complexity is made possible by

checking for wavelength availability in a parallel manner (rather than sequentially for ev-ery wavelength in every link of the path), as a result of the bitwise operations (since eachbit represents an individual wavelength and all bits are processed simultaneously duringa single ALU operation). Hence, there is no benefit to implementing this algorithm in amultiprocessor environment, since the bitwise parallel processing can be carried out in asingle processor.

The linear increase in processing time withW andL makes the SP-FF algorithm com-putationally simple and fast. However, the SP-FF algorithm results in poor blocking prob-ability performance compared with other more complex algorithms [8]. A technique toovercome this limitation of the SP-FF algorithm, while maintaining its low computationalcomplexity, is through the search of more than one path using what, in this study, is referredto as thek-SP-FF DRWA algorithm. This searches up tok disjoint shortest paths betweensource and destination; in a single-processor environment, the pseudocode above must berepeatedk times, resulting in a increased computational complexity ofO(kL+kW). How-ever, in a multiprocessor environment (k processors, one processor per path), the compu-tational complexity of thek-SP-FF algorithm would remain the same as that of the SP-FFalgorithm, while reducing the blocking probability. Hence the blocking probability can belowered at the expense of using more electronic hardware.

3.B. AUR-E Algorithm

The AUR-E algorithm, first proposed in Ref. [27], has been extensively investigated since(see, for example, Refs. [8, 12–15, 28–31]). AUR-E implements one undirected graphper wavelength, defined asGi = {V,Ei}i = 1,2, . . . , |W| whereW =

{λ1,λ2, . . . ,λ|W|

}is the set of wavelengths,V the set of nodes, andEi the set of links whereλi isnot used. When a request to establish a lightpath between source (s) and destina-tion (d) is received, the Dijkstra algorithm is executed in eachGi . As a result a setof shortest paths SPsd =

{SPsd λ1,SPsd λ2, . . . ,SPsd λ|W|

}is generated, where SPsd λi ={

lsd λi 1, lsd λi 2, . . . , lsd λi |SPsd|}

corresponds to the set of links that make up the shortestpath between source (s) and destination (d) in graphGi . From the set SPsd, the path withthe minimum number of links (hops) is chosen and the corresponding graph is updated, bydeletion of the edges corresponding to the links used in the path. On lightpath release, theselinks are again added to the graph. The implementation of the AUR-E algorithm used inthis study is as follows:


for i=1,2, ,�|W|

shortest_path[i]=Dijkstra(Gi,s,d) j=minimum_hop(shortest_path) ACK(shortest_path[j]) sent to source node

From the pseudocode above it can be seen that the asymptotic time complexity of thelightpath search task isO

(WN2 +W

), with the execution of the Dijkstra algorithm dom-

inating the computational complexity [O(N2)[32]]. Other implementations of the Dijk-

stra algorithm may yield a lower computational complexity [for example, using Fibonacciheaps instead of static arrays [33], leading toO(NlogN+L)]. However, this applies onlyto networks with a high number of nodes (� 100), not applicable to practical networks(typically less than 100 nodes). Assuming a multiprocessor environment (one processorper wavelength) the computational complexity of the AUR-E algorithm can be reduced toO(N2 +W

).

Although the Dijkstra algorithm makes the AUR-E algorithm computationally expen-sive with respect to the SP-FF algorithm and its variantk-SP-FF, the AUR-E algorithm hasbeen shown to achieve significantly better performance [8, 13–15, 29] as it searches allpossible routes (instead of a reduced set) for every request. In the remainder of this sec-tion, we investigate the trade-off between the maximum number of supported nodes andthe resulting processing time for the SP-FF and AUR-E algorithms.

For the scalability analysis of practical network architectures it is not sufficient to knowonly the asymptotic complexityO[ f (N)] of algorithms. This is because asymptotic com-plexity analysis assumes that the variables of interest (e.g.,N, L, andW used in the SP-FFand AUR-E algorithms) take very high values. Therefore, operations not involving thosevariables are considered to be executed in negligible time and are not taken into accountin the complexity analysis. In practical cases, however, the neglected operations may con-tribute significantly to the execution time. This renders asymptotic complexity analysis in-effective in accurately estimating execution times or providing tight bounds. In this study,analytically tractable upper bounds were obtained for the execution time of the studied al-gorithms using a technique known as static performance prediction [34, 35]. This techniqueconsiders all operations performed by an algorithm. The execution time of every opera-tion is then estimated from the number of memory accesses and arithmetic/logical opera-tions carried out during each operation. Because the total processing time of an algorithmdepends on the type of operations executed (dynamically chosen according to the inputdata), this technique provides an upper bound by analyzing the longest possible executiontime. Hardware- or software-dependent optimizations for speed-ups (e.g., pipelining, par-allel execution of instructions, or compiler optimizations) are not considered, since theyare specific to each implementation. As a result, the application of the static performanceprediction leads to an overestimation of the execution time (worst case).

3.C. Processing Times

Using the static performance-prediction technique, the expressions below (second columnin Table1) for the execution time of the different DRWA algorithms under investigationwere obtained. The asymptotic computational complexity is also given. The following no-tation was used throughout for the formulas in the tables (detailed derivation in Ref. [32]):

N: number of nodes

L: number of links

W: number of wavelengths per link


P̂: longest path (in number of hops)

P: mean length of paths (in number of hops)

k: number of different paths explored during the execution of thek-SP-FF algo-rithm

f ( ): a function where the coefficients are linear with memory access time (tmem)and the time required to perform an arithmetic or logical operation (tA/L).

See Ref. [32] for a detailed derivation and description of the formulas for the processingtimes of the individual algorithms, obtained using the static performance-prediction tech-nique.

Table 1. Processing Time and Computational Complexity of the Different DRWAAlgorithms (k-SP-FF and AUR-E)

Algorithm Execution Time Complexity

SP-FF single-/multiproc. f( P� ,W) O(L+W) ∼ O(N2)

k-SP-FF single proc. f(k, P� ,W) O(kL+kW) ∼ O(N2)

k-SP-FF multiproc. f ( P� ,W) O(L+W) ∼ O(N2)

AUR-E single proc. W f (N2,N,L)+ f (W, P ) O(N2W) ∼ O(N4)

AUR-E multiproc. f (N2,N,L,W, P ) O(N2) aFor the case of single- and multiprocessor implementation.

We validated the use of the static performance-prediction technique by comparing theestimated times (obtained with formulas of second column of Table1) with the measuredexecution times of the SP-FF and AUR-E algorithms for the Eurocore and NSFNet topolo-gies [1]. The results showed that the estimation is a good indication of the actual runningtimes: whereas tight bounds are provided for the AUR-E algorithm, a decrease in the realexecution time of up to 3 times can be expected for the SP-FF algorithm.

Table 2. Processing Time of Seven Arbitrarily Meshed Network Topologies for theSP-FF, 3-SP-FF, and AUR-E Algorithms

C SP-FF

Processing Time C 3-SP-FF

Processing Time C AUR-E

Network

No. Nodes

(N)

No. Links (L)

Longest Path ( P� )

No.

Wave-lengths

(W)

(single-/ multiproc.)

(µs)

(single-proc.) (µs)

(multiproc.) (µs)

(single-proc.) (ms)

(multiproc.)

(µs) USNet 46 76 11 108 1.70 3.17 1.70 4.48 43.20

Eurolarge 43 90 8 88 1.31 2.48 1.31 3.24 38.30

ARPANet 20 31 6 33 0.56 1.03 0.56 0.28 9.09

UKNet 21 39 5 21 0.42 0.76 0.42 0.20 9.87

EON 20 39 5 18 0.40 0.71 0.40 0.16 9.06

NSFNet 14 21 3 13 0.27 0.49 0.27 0.06 4.64

EuroCore 11 25 3 4 0.20 0.34 0.20 0.01 3.20 aWith the same number of wavelengths as in a static WRON and single- and multiprocessor control node architectures.


We applied the formulas given in the second column of Table1 (details of the derivationin Ref. [32]) to 7 real, arbitrarily meshed optical network topologies [1] to obtain the valuesof the maximum DRWA processing timeC for the SP-FF, 3-SP-FF (i.e.,k-SP-FF withk= 3,since a higher number of disjoint paths did not lead to significantly increased performance)and AUR-E algorithms. The maximum DRWA processing timeC was evaluated for bothsingle- and multiprocessor environments (Table2). As a worst-case assumption, the numberof wavelengths considered was equivalent to that required in the case of static lightpathassignment, whereas a lower count would be expected in dynamic networks because ofthe potential capacity savings. The processing times were calculated assuming a Pentium-4processor operating with an ALU (arithmetic logical unit) processing time of 0.83 ns, andSRAM at 1.8-GHz access speed of 1 ns [36].

By inspection of the equations for the estimate of the processing times (see detailsin Ref. [32]), it can be seen that the memory access time has a significant effect on theprocessing times. Therefore, the use of SRAM instead of DRAM is paramount owing tothe significantly lower memory access times (50 versus 1 ns, respectively). The total storagespace available in SRAM, however, is significantly lower (Mbit regime), and it is necessaryto confirm that the data structures used by the algorithms are within the SRAM capacity.For thek-SP-FF algorithm the memory requirement isO

(N3), mostly given by the size of

the routing table (kN2P̂ bytes), whereas for AUR-E the required size of memory isO(N4),

mostly determined by the structures to store the network state (WN2 bytes). The analysis tocalculate memory requirements for both algorithms (details in Ref. [32]) led to the valuesshown in Table3, and it can be seen that memory requirements are well below the limits ofcurrent SRAM designs [37].

Table 3. Memory Requirements for k-SP-FF and AUR-E Algorithms for the SameSeven Network Topologies as Used in Table3

Network 3-SP-FF (Kbyte)

AUR-E (Kbyte)

USNet 76.8 228.3 Euro large 51.6 162.9 ARPA Net 8.7 13.7 UKNet 8.4 9.6 EON 7.7 7.5 NSFNet 2.6 2.7 Euro Core 1.8 0.6

With the formulas in the second column of Table1 [32], the maximum processing timeper request,C, is plotted against the number of nodes in Fig.5 for (a) the SP-FF algorithm ina single and multiprocessor environment, as well as the case of multiprocessor implemen-tation of the 3-SP-FF algorithm; (b) the 3-SP-FF algorithm in a single-processor machine;(c) the AUR-E algorithm in a single-processor computer; and (d) the case of multiproces-sor implementation of the AUR-E algorithm. All graphs contain the results for a constantnumber of wavelengths, as well as the results for the seven network architectures listedin Table2. With the least-squares-fit method, a straight line was fitted to the processingtimes of the seven network architectures. Its slope (exponent ofN) defines the complexityof the processing time as a function of the number of nodes. It can be seen that AUR-E ina single-processor environment experiences severe computational overhead, scaling witha slope of 3.91, which can be reduced to 1.85 when a multiprocessor architecture is used.The SP-FF algorithm and the parallel implementation of the 3-SP-FF algorithm achieve the


lowest complexity, reflected in a slope of 1.46 only.

(a) (b)

(c) (d)

Fig. 5. Maximum DRWA processing timeC as a function of the number of nodes for sevenreal network architectures and using SP-FF and AUR-E algorithms in single- or multipro-cessor control node architectures: (a) single- or multiprocessor SP-FF and multiprocessor3-SP-FF, (b) single-processor 3-SP-FF, (c) single-processor AUR-E, and (d) multiprocessorAUR-E.

4. Results for Network Scalability

The results obtained in Section3 are now applied to investigate the network scalability interms of the number of nodes and for given latencies and blocking probability (QoS con-straints), as a function of the DRWA processing time for different scheduling algorithms.

4.A. Network Scalability for Operation at High Network Load

It is expected that for high traffic loads dynamic networks will not save wavelengths com-pared with static networks, since each wavelength will be highly utilized. Hence, the num-ber of wavelengths required in the case of static network operation can be considered to bean upper bound for wavelength count in the dynamic case. Figure6 shows the results ofthe achievable number of nodes as a function of the processing time when as many wave-lengths were used as in the case of static network operation. It can be seen that the AUR-E(single processor) and the SP-FF algorithm (as well as the case of multiprocessor imple-mentation of thek-SP-FF algorithm) give the lower and the upper limit to the number of


nodes, respectively. This is further bounded by the maximum allowable number of nodesfor which a given latency (edge delaytedge) can be guaranteed by the scheduling algorithmsdiscussed in Section2. The results are plotted in for the same CoS values as previously inFig. 4, with a network diameter of 1000 km and edge delays of 10, 20, and 50 ms.

It can be seen that only approximately ten nodes can be supported by a network usingthe AUR-E algorithm in a single-processor environment, which prevents the utilization ofthis algorithm for most practical networks. The parallel implementations of the DRWA al-gorithms show significant improvements in scalability over the single-processor operation:AUR-E (∼20 nodes), SP-FF/3-SP-FF (40–50 nodes depending on the diameter of the net-work). Considering a SP-FF algorithm with a factor of 3 faster execution time (accordingto our validation; see Subsection3.C), up to 50–70 nodes can then be supported. These re-sults correspond to lower bounds on the number of nodes and show that centralized opticalnetworks with centralized control can be implemented for medium-size networks.

Fig. 6. Number of nodes plotted against the DRWA processing time per requestC, forsingle- and multiprocessor SP-FF and multiprocessor 3-SP-FF (squares), single-processor3-SP-FF (circles), multiprocessor AUR-E (triangles), and single-processor AUR-E (dia-monds). The processing time is bounded by the QoS constraints, here plotted assuming thesame values as in Fig.4, with network diameters of 500 km (solid), 1000 km (dashed), and1500 km (dotted), each of which was derived for 3 CoS with edge delays of 10, 20, and 50ms.

4.B. Effect of Hardware Improvements on Network Scalability

Among the parameters affecting the DRWA algorithms’ execution time, the memory ac-cess time (tmem) and the ALU operation time (tA/L) can potentially be reduced throughimprovements in the electronic processing technology. In those cases, the number of nodessupported by a centralized architecture will be higher than those predicted by Fig.6.

Using a value oftmem ten times faster than assumed previously (i.e., 0.1 ns) leads to asignificant increase in the achievable number of nodes that can be supported, from 40–50 to55–70 for the SP-FF and the parallel implementation of the 3-SP-FF algorithms. The AUR-E algorithm exhibits a lower increase, now supporting 13–15 nodes and 24–30 nodes for thesingle- and multiprocessor environment, respectively. DecreasingtA/L instead, results in anegligible effect on the scaling for all the algorithms. This shows that faster memories havea much higher impact on scalability than faster processors [32], as previously discussed inSubsection3.C.


4.C. Network Scalability for Operation at Low Network Load

In terms of the number of wavelengths, we have until now assumed that the same networkcapacity is required by all the studied DRWA algorithms and that this capacity is equal tothe number of wavelengths required in a static network. This, however, is unlikely to be thecase, since it is expected that, at least at low loads, dynamic operation will result in wave-length savings [38–40]. Moreover, different DRWA algorithms require different networkcapacity (with varying efficiencies) for achieving the same blocking probability. In fact,the SP-FF algorithm (the highest scalable algorithm) exhibits significantly higher blockingprobability than the AUR-E algorithm. As a result, the SP-FF algorithm (and its variant 3-SP-FF) may require a much higher number of wavelengths than the AUR-E algorithm forachieving the same target blocking probability, which in turn would increase its executiontime (which increases linearly with the number of wavelengths). This highlights a complextrade-off between DRWA processing time, wavelength savings, and blocking probability,which needs to be optimized for each given topology and given input traffic matrix.

To study this trade-off, the capacity required by the SP-FF, 3-SP-FF, and AUR-E algo-rithms for achieving an average blocking probability lower than 10−4 (a value sufficientlylow to avoid performance degradation of the upper transport layers [41, 42]) has beencalculated through extensive simulation experiments where uniformON–OFF input traffic(bursts) has been assumed, withON andOFF periods exponentially distributed. The meanON period was assumed to be 5, 10, and 25 ms for UKNet, European networks (EON, Euro-core, and Eurolarge), and USA networks (NSFNET, ARPANET, and USNet), respectively.Table4 shows the required capacity per link, for a traffic load of 10% (several studies, forexample, Refs. [43, 44], have shown that nowadays real networks rarely exceed this levelof utilization) along with the capacity required in the static case.

As expected, the dynamic operation results in wavelength savings with respect to thestatic case, with the SP-FF and 3-SP-FF algorithms requiring more capacity than for AUR-E, for achieving the same blocking probability for the same traffic load (10%).

Table 4. Comparison of the Required Capacity (Number of Wavelengths) in the Caseof Static WRON (100% Load) and Using SP-FF, 3-SP-FF, and AUR-E Algorithmswith Traffic Load of 10%.

Capacity Per Link (number of wavelengths)

Network

Static Case, WRON 100%

max. Nλ (avg. Nλ per link)

Dynamic CaseSP-FF 10%

Dynamic Casek-SP-FF k=3

10%

Dynamic Case AUR-E

10% USNet 108 (59) 26 22 17 Euro large 88 (36) 20 16 11 ARPANet 33 (17) 12 9 8 UKNet 21 (13) 9 7 6 EON 18 (11) 9 7 6 NSFNet 13 (9) 7 5 5 Eurocore 4 (3) 4 3 2 aIn the case of a static WRON we distinguish between the maximum number of wavelengths in one link and the average number of wavelengths (bold, in brackets).


Table 5. Processing Time of Seven Arbitrarily Meshed Network Topologies for theSP-FF, 3-SP-FF, and AUR-E Algorithms

Process. Time C 3-SP-FF

Process. Time C AUR-E

Network

Process. Time C

SP-FF (single-

/multiproc.) (µs)

(singleproc.)(µs)

(multiproc.)(µs)

(single-proc.) (µs)

(multiproc.) (µs)

USNet 0.81 1.29 0.74 705 42.5 Eurolarge 0.59 0.94 0.54 440 37.4 ARPANet 0.40 0.63 0.38 67.8 8.76 UKNet 0.33 0.54 0.32 56.7 9.69 EON 0.33 0.54 0.32 52.1 8.93 NSFNet 0.23 0.35 0.21 22.0 4.53 EuroCore 0.20 0.32 0.20 6.22 3.19 aWith the number of wavelengths required for achieving a blocking probability

of 410− and for a traffic load of 10%.

Table5 shows the corresponding processing times with the new considered wavelengthcount for reduced load. From data in Tables4 and5 a trade-off between processing time andblocking (capacity requirements) of the analyzed algorithms can be observed: whereas SP-FF is the fastest algorithm (Table5), it requires the highest capacity (Table4) for achievingthe target blocking probability. Conversely, AUR-E is the slowest algorithm but achievesthe lowest capacity requirement. This trade-off, however, is based only on the three ana-lyzed DRWA algorithms. Other DRWA algorithms may not exhibit a compromise betweenprocessing time and blocking. For example, shortest-path random-fit (SP-RF) has a higherprocessing time than SP-FF but also a higher blocking. Also, potentially new algorithmsfor achieving blocking that is as good as the best but at reduced computational complexitywould not suffer from this trade-off.

Applying a linear fit to the data of Table5 and using the same technique as applied inFig. 6, we quantified the scalability of the algorithms, considering the wavelength count re-duction. Although significant in most cases, the effect of the wavelength count reduction onthe scalability is lower than a decrease in the memory access times. The number of nodessupported by the AUR-E algorithm in a multiprocessor environment remains virtually un-changed (with one processor per wavelength, the processing time is mostly insensitive tothe number of wavelengths), whereas the scaling of the 3-SP-FF algorithm improves withthe wavelength reduction, leading to an increase from 40–50 to 50–65 nodes. These resultslead to the following recommendations on the choice of DRWA algorithm in a centralizednetwork:

1. The SP-FF algorithm requires significant overprovisioning of wavelengths (with re-spect to the other dynamic alternatives) for achieving an acceptable blocking prob-ability, and, since the parallel implementation of thek-SP-FF algorithm requiresreduced resources with the same computational complexity, the SP-FF algorithmshould not be used in these applications.

2. Although it requires significantly lower resources, the AUR-E algorithm imple-mented in a single-processor core node allows only networks with low number of


nodes (approximately 10). A parallel version of the algorithm is more suitable forimplementation in real networks, leading to a node count of roughly 20.

3. The choice between thek-SP-FF and AUR-E algorithm implemented in a multipro-cessor environment will be determined by the size of the network, the maximumnumber of nodes supported by the DRWA algorithms, and thecostof implement-ing thek-SP-FF or AUR-E algorithm. LetN andNk-SP-FF (NAUR-E) be the numberof nodes of the network and the maximum number of nodes supported by a centralnode executing thek-SP-FF (AUR-E) algorithm, respectively. From results obtainedhere,Nk-SP-FF> NAUR-E. Then, if

• N > Nk-SP-FF, a centralized dynamic implementation is not possible;

• NAUR-E < N < Nk-SP-FF, k-SP-FF should be used;

• N < NAUR-E < Nk-SP-FF, either thek-SP-FF or AUR-E algorithm can supportthe network.

In the final case (N < NAUR-E < Nk-SP-FF) the choice of the DRWA algorithm willbe determined by the cost of its implementation. Assuming that cost is determinedmostly by the number of processors (k andW processors required for thek-SP-FF andAUR-E algorithm, respectively) and required network capacity (line card plus opticaltransmitter/receiver per wavelength), it is then possible to select the algorithm thatminimizes the cost for a given network architecture.

5. Summary and Conclusions

Lower bounds on the maximum number of nodes supported by centralized dynamic opticalnetworks with end-to-end lightpath assignment were obtained as a function of the process-ing of lightpath requests in the control node. Processing in the control node is achieved byscheduling algorithms (FIFO, rate-monotonic, and earliest-deadline-first) and by the exe-cution of DRWA algorithms (k-SP-FF and AUR-E). The worst case in terms of networkscalability was investigated by assuming the longest execution times of sequential process-ing of incoming requests arriving at the maximum rate for which the scheduling algorithms(RM and EDF algorithms) can guarantee fairness. The electronic implementation of theDRWA algorithms studied was also investigated, with a focus on the effectiveness of par-allel processing to minimize the processing time per request and maximize the achievablenumber of nodes to be supported by the network architecture for given QoS constraints(latency and blocking). During operation with the wavelength count required in the staticcase (considered an upper bound for the network capacity here), it was shown that usingthe SP-FF algorithm (and the parallel implementation of 3-SP-FF) in the control node canachieve a lower bound of 40–50 nodes, whereas the AUR-E algorithm can only support upto 10 and 20 nodes in the cases of single- and multiprocessor environments, respectively. Itwas shown that a reduction in the memory access times increases the scalability (by∼30%in the case of the SP-FF algorithm), whereas increasing the processor speed has a negligibleeffect.

The study further showed that there is a trade-off between processing time, blockingprobability, and resource requirements for the investigated DRWA algorithms, with the SP-FF algorithm typically operating more than 100 times faster, but experiencing blockingprobabilities that are 100 times higher than with the AUR-E algorithm. This translates intosignificant resource requirements for the SP-FF algorithm compared with AUR-E, whichin turn affects its processing time. It was found that the parallel implementationk-SP-FFprovides the best compromise between performance and scalability, with a scalability ofup to 70 nodes for 1500-km-diameter networks operating at traffic load of 10% (50 nodes


for the same capacity required in the static case), showing that even under stringent QoSconstraints (edge delay, 10 ms) and worst-case processing assumptions, centralized supportof the maximum number of nodes in present networks can be achieved. Future research ofadvanced DRWA algorithms in centralized architectures should aim at investigating thetrade-off between blocking probability and processing time.

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investigation of the scalability of dynamic wavelength-routed optical networks [invited]

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